Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11939 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,19,24,17,22,32,23,20,25,35,40,33,41,18,34)(16,26,29,38,31,42,45,36,30,43,44,21,27,37,28,39);(6,7,9,11)(8,13,12,10)(14,16,20,30)(15,21,25,38)(17,27,33,31)(18,29,32,44)(19,28,35,45)(22,36,41,39)(23,42,34,37)(24,43,40,26);(14,17,20,33)(15,22,25,41)(16,27,30,31)(18,19,32,35)(21,36,38,39)(23,40,34,24)(26,37,43,42)(28,44,45,29)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4927);